The automorphism group of a finite p-group is almost always a p-group

JOURNAL OF ALGEBRA

卷:312

Article

Helleloid, Geir T. ; Martin, Ursula

关键词: p-group;    automorphism group;    lower p-series;    Frattini subgroup;    lower central p-series;   

DOI  :  10.1016/j.jalgebra.2007.01.008

来源: Elsevier

【 摘 要 】

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups whose automorphism group is a p-group. Yet the goal of this paper is to prove that the automorphism group of a finite p-group is almost always a p-group. The asymptotics in our theorem involve fixing any two of the following parameters and letting the third go to infinity: the lower p-length, the number of generators, and p. The proof of this theorem depends on a variety of topics: counting subgroups of a p-group; analyzing the lower p-series of a free group via its connection with the free Lie algebra; counting submodules of a module via Hall polynomials; and using numerical estimates on Gaussian coefficients. (C) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free